Robinson, Christopher Alan (1969) On the enumeration of cross-sections and unstable vector bundles. PhD thesis, University of Warwick.

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Abstract

This thesis is an attempt to generalise the methods of two papers [9], [11] by James and Thomas on enumeration problems.

The first five sections consider the problem of enumerating the homotopy classes of cross-sections of a fibration F c X p B. In §l, we define certain groups no*(p) by stabilizing (in a suitable sense) the homotopy groups of the space of cross-sections of p. We obtain generalised 'difference classes' and 'obstruction classes' as elements of noo (p) and no -1 (p) respectively, and hence relate
these to the enumeration problem.

§2 develops the theory of modified Postnikov towers (MPT's) for fibrations in which the fundamental group of the base acts non-trivially. For this we need techniques for handling k-invariants with local coefficients, but the theory in other respects parallels that of Mahowald [13] and Thomas [23]. An existence theorem is proved in 2.10. §3 introduces the notion of 'stable modified Postnikov tower', which is a crude device to facilitate the setting
up of the spectral sequence in the following section. A stable MPT for a fibration kills the stable homotopy groups of the fibre, just as an MPT kills the unstable ones. The existence of stable towers follows from the possibility of 'de-looping' MPT's in the stable range (3.3). In §4 we set up a spectral sequence (4.19) which, in the simplest case, has the form E~,t ~ Ht(B;rrEsF) => rr: (p) where F is the s'th stable homotopy group of F. This is our main tool for calculating rr: (p) , and for enumerating cross-sections. An essentially similar approach to the enumeration problem has been used (independently) by J.F. lTcClendon [12]. Our spectral sequence appears to be a formalisation of the last section of [12].

In §5 we compute MPT's for the fibrations BOn --> BO. These extend the calculations of Mahowald [13] for BSOn --> BSO to the non-orientable case and to the next two homotopy groups of the fibre. Instead of giving the defining relations of the k-invariants, we display the corresponding differentials in our spectral sequence: these contain equivalent information. We apply the results of the computations to determine the number of regular homotopy classes of immersions of real projective n-space Pn in Euclidean space in dimensions near the stable range.

§6 considers the problem of determining the number of real k-plane bundles over a complex X which are stably equivalent to a given bundle. The methods here are developments of those of James and Thomas [9]. We generalise one of the theorems of [9] to nonorientable
bundles, and enumerate (n-l)- and (n-2)- dimensional normal bundles to Pn in certain cases.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Combinatorial enumeration problems, Homotopy groups, Vector bundles
Official Date:	1969
Dates:	Date Event 1969 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Epstein, D. B. A.
Extent:	110 leaves.
Language:	eng

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